plot.ts(var_ts1 , main = "", xlab = "")
ARIMAX/SARIMAX/VAR
Summary
In the previous modeling sections, we analyzed a univariate time series of monthly inflation rates in the United States, calculated from the Consumer Price Index (CPI) data spanning several decades. While ARIMA and SARIMA models provided insights into the inflation trends and seasonal patterns, we aim to enhance our understanding by incorporating endogenous variables into our analysis. Endogenous variables in the context of inflation might include factors like Disposable Income, unemployment rates, Personal Consumption, and major economic policies or events. These variables are determined within the economic system and are influenced by other variables in the system, often demonstrating interdependence and reacting to changes in other economic indicators.
Literature Review
Numerous studies have explored the relationship between macroeconomic indicators and inflation. A study by Sims (1980) introduced the Vector Autoregression (VAR) approach, which has been widely used in economic forecasting and policy analysis. This approach allows for the modeling of the dynamic interaction among multiple macroeconomic variables. Bernanke et al. (1998) further refined the VAR approach by incorporating structural analysis, providing deeper insights into how monetary policy affects the economy, including inflation.
Research by Stock and Watson (1999) demonstrated the predictive power of incorporating multiple macroeconomic variables in forecasting inflation, showing that models with more variables tend to outperform simpler univariate models. This highlights the importance of considering a broad range of economic indicators to understand inflation dynamics more comprehensively.
VAR Model Justification
A VAR model is preferred over simpler ARIMAX models in this context for several reasons:
Simultaneous Modeling of Multiple Time Series: VAR models can simultaneously account for multiple interdependent economic indicators, capturing the complex relationships between them.
Handling Lags Efficiently: VAR models can effectively handle multiple lags in economic data, which is crucial in understanding the delayed effects of economic policies and events on inflation.
Robustness to Missing Data: Given the historical nature of economic data, VAR models can handle instances of missing data more efficiently than ARIMAX models.
Dynamic Relationships: VAR models are well-suited to capture the evolving relationships between economic variables over time, which is essential in understanding how various factors influence inflation.
Methodology
Our analysis will be divided into two parts, using both VAR and ARIMAX models. The first part will forecast monthly inflation rates using historical CPI data, considering various macroeconomic variables as potential predictors. The second part will focus on a more recent period to capture the effects of contemporary economic policies and global events on inflation.
We will also differentiate between VAR and ARIMAX models based on their capacity to handle multiple variables. While the VAR models will consider a broad range of economic indicators, the ARIMAX models will focus on specific variables identified as key influencers of inflation, such as interest rates and unemployment rates.
This approach will enable us to determine the relative impact of different economic factors on inflation and improve our ability to forecast future inflation trends based on current economic conditions.
Time Series Plot
Pair Plot
# create scatterplot matrix using plotly
fig <- plot_ly(
data = as.data.frame(var_ts1),
type = "splom",
diagonal = list(visible = FALSE),
dimensions = list(
list(label = "CPI", values = ~CPIAUCSL),
list(label = "Unemployment", values = ~UNRATE),
list(label = "DisposableIncome", values = ~DSPIC96),
list(label = "PersonalConsumption", values = ~PCE),
list(label = "TreasuryYield", values = ~GS10),
list(label = "FederalFundsRate", values = ~FEDFUNDS)
)
) %>%
layout(hovermode = "x")
# customize layout
fig <- fig %>%
layout(
title = "Scatterplot Matrix of VAR Model Variables",
xaxis = list(title = ""),
yaxis = list(title = "")
)
# display plot
fig# convert the matrix to a time series object with a yearly frequency
var_ts1 <- ts(var_ts1, frequency = 1,
start = 1970)
# split into train and test sets
set.seed(29830)
train_idx <- sample(nrow(var_ts1), 0.9 * nrow(var_ts1))
train <- var_ts1[train_idx, ]
test <- var_ts1[-train_idx, ]Fitting the VAR Model
Here we use the VARselect() function to find the best p to fit VAR(p). We will choose a maximum lag of 10 and check which p value returns lowest AIC.
(var_result <- VARselect(var_ts1, lag.max = 10, type = "both"))$selection
AIC(n) HQ(n) SC(n) FPE(n)
5 3 2 5
$criteria
1 2 3 4 5 6
AIC(n) 5.750435 5.112869 5.006714 4.957243 4.953009 4.981328
HQ(n) 5.889067 5.355475 5.353294 5.407796 5.507537 5.639829
SC(n) 6.106319 5.735665 5.896423 6.113864 6.376543 6.671775
FPE(n) 314.330622 166.155229 149.436154 142.249649 141.690540 145.823030
7 8 9 10
AIC(n) 4.976799 5.018745 5.021673 5.058962
HQ(n) 5.739275 5.885194 5.992096 6.133359
SC(n) 6.934159 7.243017 7.512858 7.817059
FPE(n) 145.249853 151.589710 152.184594 158.160557Now, we will fit VAR(1), VAR(2), and VAR(3):
VAR(1) output:
summary(fit <- VAR(var_ts1, p=1, type="both"))
VAR Estimation Results:
=========================
Endogenous variables: CPIAUCSL, UNRATE, DSPIC96, PCE, GS10, FEDFUNDS
Deterministic variables: both
Sample size: 600
Log Likelihood: -6769.181
Roots of the characteristic polynomial:
0.9993 0.9868 0.9823 0.9823 0.9081 0.9081
Call:
VAR(y = var_ts1, p = 1, type = "both")
Estimation results for equation CPIAUCSL:
=========================================
CPIAUCSL = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 9.808e-01 6.108e-03 160.576 < 2e-16 ***
UNRATE.l1 2.348e-02 1.366e-02 1.719 0.08621 .
DSPIC96.l1 1.807e-04 1.080e-04 1.673 0.09480 .
PCE.l1 -1.330e-04 5.939e-05 -2.240 0.02546 *
GS10.l1 -2.084e-02 2.026e-02 -1.028 0.30417
FEDFUNDS.l1 5.960e-02 1.268e-02 4.700 3.24e-06 ***
const -2.246e-01 4.949e-01 -0.454 0.65011
trend 7.612e-03 2.933e-03 2.595 0.00968 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4126 on 592 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 2.193e+06 on 7 and 592 DF, p-value: < 2.2e-16
Estimation results for equation UNRATE:
=======================================
UNRATE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.187e-02 2.509e-03 4.730 2.81e-06 ***
UNRATE.l1 1.002e+00 5.612e-03 178.637 < 2e-16 ***
DSPIC96.l1 1.674e-04 4.435e-05 3.775 0.000176 ***
PCE.l1 -6.604e-05 2.440e-05 -2.707 0.006986 **
GS10.l1 -2.182e-02 8.322e-03 -2.621 0.008983 **
FEDFUNDS.l1 2.023e-02 5.209e-03 3.884 0.000114 ***
const -9.091e-01 2.033e-01 -4.472 9.29e-06 ***
trend -6.327e-03 1.205e-03 -5.252 2.11e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1695 on 592 degrees of freedom
Multiple R-Squared: 0.9888, Adjusted R-squared: 0.9887
F-statistic: 7487 on 7 and 592 DF, p-value: < 2.2e-16
Estimation results for equation DSPIC96:
========================================
DSPIC96 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -1.863527 1.011445 -1.842 0.06591 .
UNRATE.l1 -7.201908 2.262177 -3.184 0.00153 **
DSPIC96.l1 0.895838 0.017879 50.106 < 2e-16 ***
PCE.l1 0.054053 0.009834 5.496 5.77e-08 ***
GS10.l1 3.917953 3.354882 1.168 0.24334
FEDFUNDS.l1 -3.606037 2.099888 -1.717 0.08646 .
const 474.101366 81.949981 5.785 1.17e-08 ***
trend 1.517586 0.485628 3.125 0.00187 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 68.33 on 592 degrees of freedom
Multiple R-Squared: 0.9996, Adjusted R-squared: 0.9996
F-statistic: 2.212e+05 on 7 and 592 DF, p-value: < 2.2e-16
Estimation results for equation PCE:
====================================
PCE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -1.369810 0.398034 -3.441 0.000619 ***
UNRATE.l1 -0.904493 0.890235 -1.016 0.310037
DSPIC96.l1 0.004132 0.007036 0.587 0.557198
PCE.l1 0.996026 0.003870 257.369 < 2e-16 ***
GS10.l1 1.687626 1.320248 1.278 0.201657
FEDFUNDS.l1 -0.214929 0.826369 -0.260 0.794886
const 24.072882 32.249800 0.746 0.455692
trend 0.612548 0.191109 3.205 0.001422 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 26.89 on 592 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 2.049e+06 on 7 and 592 DF, p-value: < 2.2e-16
Estimation results for equation GS10:
=====================================
GS10 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -1.170e-02 4.351e-03 -2.690 0.00734 **
UNRATE.l1 9.329e-03 9.731e-03 0.959 0.33810
DSPIC96.l1 -1.479e-04 7.691e-05 -1.923 0.05498 .
PCE.l1 3.879e-05 4.230e-05 0.917 0.35958
GS10.l1 9.608e-01 1.443e-02 66.582 < 2e-16 ***
FEDFUNDS.l1 2.072e-02 9.033e-03 2.293 0.02217 *
const 9.874e-01 3.525e-01 2.801 0.00526 **
trend 6.247e-03 2.089e-03 2.990 0.00290 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2939 on 592 degrees of freedom
Multiple R-Squared: 0.9909, Adjusted R-squared: 0.9908
F-statistic: 9169 on 7 and 592 DF, p-value: < 2.2e-16
Estimation results for equation FEDFUNDS:
=========================================
FEDFUNDS = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -3.060e-02 7.902e-03 -3.873 0.000120 ***
UNRATE.l1 -4.754e-02 1.767e-02 -2.690 0.007351 **
DSPIC96.l1 -1.187e-04 1.397e-04 -0.850 0.395625
PCE.l1 5.642e-05 7.683e-05 0.734 0.463016
GS10.l1 1.016e-01 2.621e-02 3.875 0.000119 ***
FEDFUNDS.l1 9.292e-01 1.640e-02 56.643 < 2e-16 ***
const 1.372e+00 6.402e-01 2.143 0.032557 *
trend 1.280e-02 3.794e-03 3.375 0.000787 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5338 on 592 degrees of freedom
Multiple R-Squared: 0.9816, Adjusted R-squared: 0.9814
F-statistic: 4524 on 7 and 592 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 0.170259 -0.005113 -5.13816 3.9249 0.021823 0.01893
UNRATE -0.005113 0.028730 0.03106 -0.4434 -0.008505 -0.01666
DSPIC96 -5.138156 0.031063 4668.55214 -37.0247 -0.429415 -0.71821
PCE 3.924950 -0.443416 -37.02471 723.0004 0.555891 0.57875
GS10 0.021823 -0.008505 -0.42941 0.5559 0.086381 0.05497
FEDFUNDS 0.018928 -0.016662 -0.71821 0.5787 0.054966 0.28493
Correlation matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 1.00000 -0.073101 -0.182247 0.35376 0.17995 0.08594
UNRATE -0.07310 1.000000 0.002682 -0.09729 -0.17072 -0.18416
DSPIC96 -0.18225 0.002682 1.000000 -0.02015 -0.02138 -0.01969
PCE 0.35376 -0.097292 -0.020153 1.00000 0.07034 0.04032
GS10 0.17995 -0.170723 -0.021383 0.07034 1.00000 0.35036
FEDFUNDS 0.08594 -0.184156 -0.019692 0.04032 0.35036 1.00000VAR(2) output:
summary(fit <- VAR(var_ts1, p=2, type="both"))
VAR Estimation Results:
=========================
Endogenous variables: CPIAUCSL, UNRATE, DSPIC96, PCE, GS10, FEDFUNDS
Deterministic variables: both
Sample size: 599
Log Likelihood: -6533.712
Roots of the characteristic polynomial:
1 0.9846 0.9683 0.9683 0.9149 0.876 0.4178 0.3643 0.3643 0.2837 0.2198 0.1095
Call:
VAR(y = var_ts1, p = 2, type = "both")
Estimation results for equation CPIAUCSL:
=========================================
CPIAUCSL = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.332e+00 4.079e-02 32.656 < 2e-16 ***
UNRATE.l1 7.847e-02 9.250e-02 0.848 0.396587
DSPIC96.l1 1.502e-04 2.279e-04 0.659 0.509913
PCE.l1 2.146e-03 6.093e-04 3.522 0.000462 ***
GS10.l1 6.798e-02 5.645e-02 1.204 0.228938
FEDFUNDS.l1 6.398e-02 3.078e-02 2.079 0.038096 *
CPIAUCSL.l2 -3.472e-01 4.074e-02 -8.523 < 2e-16 ***
UNRATE.l2 -4.980e-02 9.339e-02 -0.533 0.594023
DSPIC96.l2 6.233e-05 2.281e-04 0.273 0.784786
PCE.l2 -2.300e-03 6.101e-04 -3.770 0.000180 ***
GS10.l2 -8.981e-02 5.630e-02 -1.595 0.111173
FEDFUNDS.l2 -2.116e-02 3.080e-02 -0.687 0.492337
const -4.592e-01 4.647e-01 -0.988 0.323472
trend 5.600e-03 2.759e-03 2.029 0.042868 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3709 on 585 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 1.455e+06 on 13 and 585 DF, p-value: < 2.2e-16
Estimation results for equation UNRATE:
=======================================
UNRATE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.129e-02 1.822e-02 0.620 0.535767
UNRATE.l1 1.049e+00 4.133e-02 25.382 < 2e-16 ***
DSPIC96.l1 7.986e-05 1.018e-04 0.784 0.433185
PCE.l1 -1.136e-03 2.722e-04 -4.174 3.44e-05 ***
GS10.l1 -4.383e-02 2.522e-02 -1.738 0.082768 .
FEDFUNDS.l1 -1.212e-02 1.375e-02 -0.881 0.378787
CPIAUCSL.l2 -2.694e-03 1.820e-02 -0.148 0.882412
UNRATE.l2 -4.968e-02 4.173e-02 -1.191 0.234303
DSPIC96.l2 6.881e-05 1.019e-04 0.675 0.499942
PCE.l2 1.078e-03 2.726e-04 3.952 8.68e-05 ***
GS10.l2 2.662e-02 2.516e-02 1.058 0.290402
FEDFUNDS.l2 3.132e-02 1.376e-02 2.276 0.023214 *
const -7.552e-01 2.077e-01 -3.637 0.000300 ***
trend -4.764e-03 1.233e-03 -3.864 0.000124 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1658 on 585 degrees of freedom
Multiple R-Squared: 0.9894, Adjusted R-squared: 0.9892
F-statistic: 4207 on 13 and 585 DF, p-value: < 2.2e-16
Estimation results for equation DSPIC96:
========================================
DSPIC96 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -21.71041 7.38894 -2.938 0.00343 **
UNRATE.l1 -28.75670 16.75780 -1.716 0.08669 .
DSPIC96.l1 0.72208 0.04128 17.492 < 2e-16 ***
PCE.l1 0.11559 0.11038 1.047 0.29544
GS10.l1 9.94580 10.22646 0.973 0.33118
FEDFUNDS.l1 -4.26814 5.57660 -0.765 0.44436
CPIAUCSL.l2 20.27753 7.38086 2.747 0.00619 **
UNRATE.l2 21.98608 16.91809 1.300 0.19426
DSPIC96.l2 0.19711 0.04133 4.769 2.34e-06 ***
PCE.l2 -0.07270 0.11053 -0.658 0.51100
GS10.l2 -5.92789 10.19874 -0.581 0.56130
FEDFUNDS.l2 1.70186 5.58020 0.305 0.76049
const 375.76035 84.18901 4.463 9.68e-06 ***
trend 1.17311 0.49986 2.347 0.01926 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 67.2 on 585 degrees of freedom
Multiple R-Squared: 0.9996, Adjusted R-squared: 0.9996
F-statistic: 1.227e+05 on 13 and 585 DF, p-value: < 2.2e-16
Estimation results for equation PCE:
====================================
PCE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 13.895207 2.877179 4.829 1.75e-06 ***
UNRATE.l1 -17.149924 6.525321 -2.628 0.008809 **
DSPIC96.l1 0.006826 0.016075 0.425 0.671268
PCE.l1 0.831833 0.042980 19.354 < 2e-16 ***
GS10.l1 -1.455403 3.982084 -0.365 0.714879
FEDFUNDS.l1 -2.117331 2.171472 -0.975 0.329930
CPIAUCSL.l2 -15.358472 2.874035 -5.344 1.31e-07 ***
UNRATE.l2 16.235457 6.587737 2.464 0.014007 *
DSPIC96.l2 -0.001770 0.016093 -0.110 0.912473
PCE.l2 0.163794 0.043040 3.806 0.000156 ***
GS10.l2 3.266601 3.971290 0.823 0.411097
FEDFUNDS.l2 1.485204 2.172876 0.684 0.494549
const 22.615719 32.782368 0.690 0.490547
trend 0.642373 0.194642 3.300 0.001025 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 26.17 on 585 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 1.162e+06 on 13 and 585 DF, p-value: < 2.2e-16
Estimation results for equation GS10:
=====================================
GS10 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 4.145e-02 3.020e-02 1.373 0.1704
UNRATE.l1 -1.361e-01 6.849e-02 -1.987 0.0473 *
DSPIC96.l1 2.064e-05 1.687e-04 0.122 0.9027
PCE.l1 1.127e-03 4.511e-04 2.499 0.0127 *
GS10.l1 1.251e+00 4.180e-02 29.923 < 2e-16 ***
FEDFUNDS.l1 -1.862e-02 2.279e-02 -0.817 0.4142
CPIAUCSL.l2 -4.634e-02 3.017e-02 -1.536 0.1251
UNRATE.l2 1.554e-01 6.915e-02 2.247 0.0250 *
DSPIC96.l2 -1.173e-04 1.689e-04 -0.695 0.4875
PCE.l2 -1.120e-03 4.518e-04 -2.478 0.0135 *
GS10.l2 -3.150e-01 4.168e-02 -7.557 1.6e-13 ***
FEDFUNDS.l2 4.984e-02 2.281e-02 2.185 0.0293 *
const 6.527e-01 3.441e-01 1.897 0.0583 .
trend 3.135e-03 2.043e-03 1.535 0.1254
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2747 on 585 degrees of freedom
Multiple R-Squared: 0.9921, Adjusted R-squared: 0.9919
F-statistic: 5659 on 13 and 585 DF, p-value: < 2.2e-16
Estimation results for equation FEDFUNDS:
=========================================
FEDFUNDS = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -0.0456235 0.0510978 -0.893 0.372296
UNRATE.l1 -0.4759622 0.1158875 -4.107 4.58e-05 ***
DSPIC96.l1 -0.0001574 0.0002855 -0.551 0.581543
PCE.l1 0.0002825 0.0007633 0.370 0.711414
GS10.l1 0.5268645 0.0707205 7.450 3.37e-13 ***
FEDFUNDS.l1 1.2355455 0.0385646 32.038 < 2e-16 ***
CPIAUCSL.l2 0.0359225 0.0510419 0.704 0.481847
UNRATE.l2 0.4448581 0.1169960 3.802 0.000158 ***
DSPIC96.l2 0.0002183 0.0002858 0.764 0.445329
PCE.l2 -0.0003050 0.0007644 -0.399 0.690043
GS10.l2 -0.4494898 0.0705288 -6.373 3.76e-10 ***
FEDFUNDS.l2 -0.3013560 0.0385896 -7.809 2.67e-14 ***
const 0.2013182 0.5822041 0.346 0.729628
trend 0.0029051 0.0034568 0.840 0.401020
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4647 on 585 degrees of freedom
Multiple R-Squared: 0.9862, Adjusted R-squared: 0.9859
F-statistic: 3223 on 13 and 585 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 0.137601 -0.001106 -4.549e+00 3.4120 0.010362 0.007159
UNRATE -0.001106 0.027474 -1.712e-03 -0.5035 -0.005972 -0.009845
DSPIC96 -4.549058 -0.001712 4.516e+03 -14.9202 -0.370402 -1.288496
PCE 3.412025 -0.503523 -1.492e+01 684.7491 0.457156 0.524037
GS10 0.010362 -0.005972 -3.704e-01 0.4572 0.075443 0.038994
FEDFUNDS 0.007159 -0.009845 -1.288e+00 0.5240 0.038994 0.215974
Correlation matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 1.00000 -0.0179823 -0.1824864 0.351509 0.10170 0.04153
UNRATE -0.01798 1.0000000 -0.0001537 -0.116089 -0.13118 -0.12781
DSPIC96 -0.18249 -0.0001537 1.0000000 -0.008485 -0.02007 -0.04126
PCE 0.35151 -0.1160891 -0.0084845 1.000000 0.06360 0.04309
GS10 0.10170 -0.1311763 -0.0200670 0.063605 1.00000 0.30548
FEDFUNDS 0.04153 -0.1278084 -0.0412574 0.043092 0.30548 1.00000VAR(3) output:
summary(fit <- VAR(var_ts1, p=3, type="both"))
VAR Estimation Results:
=========================
Endogenous variables: CPIAUCSL, UNRATE, DSPIC96, PCE, GS10, FEDFUNDS
Deterministic variables: both
Sample size: 598
Log Likelihood: -6455.747
Roots of the characteristic polynomial:
1 0.9822 0.9737 0.9737 0.9449 0.9028 0.5495 0.4541 0.4541 0.4277 0.4277 0.4055 0.3394 0.3394 0.3173 0.3173 0.223 0.223
Call:
VAR(y = var_ts1, p = 3, type = "both")
Estimation results for equation CPIAUCSL:
=========================================
CPIAUCSL = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.3999529 0.0442577 31.632 < 2e-16 ***
UNRATE.l1 0.0972676 0.0924184 1.052 0.293024
DSPIC96.l1 0.0002741 0.0002283 1.201 0.230377
PCE.l1 0.0023667 0.0006194 3.821 0.000147 ***
GS10.l1 0.0882201 0.0579665 1.522 0.128577
FEDFUNDS.l1 0.0627856 0.0341177 1.840 0.066242 .
CPIAUCSL.l2 -0.5854177 0.0684994 -8.546 < 2e-16 ***
UNRATE.l2 -0.0386682 0.1317426 -0.294 0.769235
DSPIC96.l2 0.0005502 0.0002792 1.971 0.049206 *
PCE.l2 -0.0018320 0.0007718 -2.373 0.017947 *
GS10.l2 -0.1320624 0.0883875 -1.494 0.135687
FEDFUNDS.l2 -0.0014276 0.0521380 -0.027 0.978165
CPIAUCSL.l3 0.1717438 0.0426604 4.026 6.43e-05 ***
UNRATE.l3 -0.0290749 0.0934086 -0.311 0.755710
DSPIC96.l3 -0.0006750 0.0002289 -2.950 0.003310 **
PCE.l3 -0.0006551 0.0006330 -1.035 0.301204
GS10.l3 0.0169111 0.0588071 0.288 0.773780
FEDFUNDS.l3 -0.0133484 0.0323942 -0.412 0.680448
const -0.2750833 0.4670350 -0.589 0.556092
trend 0.0053832 0.0027764 1.939 0.053000 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3633 on 578 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 1.033e+06 on 19 and 578 DF, p-value: < 2.2e-16
Estimation results for equation UNRATE:
=======================================
UNRATE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 2.605e-02 1.973e-02 1.320 0.18722
UNRATE.l1 1.016e+00 4.120e-02 24.659 < 2e-16 ***
DSPIC96.l1 9.791e-05 1.018e-04 0.962 0.33655
PCE.l1 -1.204e-03 2.761e-04 -4.361 1.53e-05 ***
GS10.l1 -4.285e-02 2.584e-02 -1.658 0.09781 .
FEDFUNDS.l1 1.047e-02 1.521e-02 0.688 0.49144
CPIAUCSL.l2 -2.436e-02 3.054e-02 -0.798 0.42533
UNRATE.l2 1.524e-01 5.873e-02 2.596 0.00968 **
DSPIC96.l2 -5.697e-05 1.244e-04 -0.458 0.64728
PCE.l2 5.165e-04 3.441e-04 1.501 0.13393
GS10.l2 4.085e-02 3.941e-02 1.037 0.30034
FEDFUNDS.l2 -2.391e-02 2.324e-02 -1.029 0.30406
CPIAUCSL.l3 4.860e-03 1.902e-02 0.256 0.79840
UNRATE.l3 -1.724e-01 4.164e-02 -4.140 3.99e-05 ***
DSPIC96.l3 8.292e-05 1.020e-04 0.813 0.41673
PCE.l3 6.388e-04 2.822e-04 2.264 0.02397 *
GS10.l3 -1.217e-02 2.622e-02 -0.464 0.64274
FEDFUNDS.l3 3.125e-02 1.444e-02 2.164 0.03088 *
const -6.055e-01 2.082e-01 -2.908 0.00378 **
trend -3.654e-03 1.238e-03 -2.952 0.00329 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.162 on 578 degrees of freedom
Multiple R-Squared: 0.99, Adjusted R-squared: 0.9897
F-statistic: 3010 on 19 and 578 DF, p-value: < 2.2e-16
Estimation results for equation DSPIC96:
========================================
DSPIC96 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -29.12109 8.10177 -3.594 0.000353 ***
UNRATE.l1 -30.02520 16.91801 -1.775 0.076466 .
DSPIC96.l1 0.69196 0.04180 16.555 < 2e-16 ***
PCE.l1 0.11451 0.11338 1.010 0.312922
GS10.l1 11.79430 10.61128 1.111 0.266821
FEDFUNDS.l1 -6.19385 6.24555 -0.992 0.321749
CPIAUCSL.l2 37.95807 12.53942 3.027 0.002579 **
UNRATE.l2 24.42613 24.11665 1.013 0.311564
DSPIC96.l2 0.08208 0.05110 1.606 0.108756
PCE.l2 -0.09091 0.14129 -0.643 0.520201
GS10.l2 -8.52430 16.18011 -0.527 0.598508
FEDFUNDS.l2 6.96975 9.54433 0.730 0.465533
CPIAUCSL.l3 -10.35743 7.80936 -1.326 0.185269
UNRATE.l3 -1.64514 17.09927 -0.096 0.923386
DSPIC96.l3 0.15822 0.04189 3.777 0.000175 ***
PCE.l3 0.01317 0.11588 0.114 0.909542
GS10.l3 2.80167 10.76517 0.260 0.794761
FEDFUNDS.l3 -4.57569 5.93006 -0.772 0.440660
const 329.61650 85.49492 3.855 0.000129 ***
trend 1.10550 0.50825 2.175 0.030025 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 66.5 on 578 degrees of freedom
Multiple R-Squared: 0.9996, Adjusted R-squared: 0.9996
F-statistic: 8.545e+04 on 19 and 578 DF, p-value: < 2.2e-16
Estimation results for equation PCE:
====================================
PCE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 16.05085 3.17036 5.063 5.56e-07 ***
UNRATE.l1 -16.36366 6.62032 -2.472 0.013733 *
DSPIC96.l1 0.01171 0.01636 0.716 0.474330
PCE.l1 0.82297 0.04437 18.549 < 2e-16 ***
GS10.l1 -0.32174 4.15238 -0.077 0.938265
FEDFUNDS.l1 -1.06643 2.44400 -0.436 0.662749
CPIAUCSL.l2 -20.50444 4.90690 -4.179 3.39e-05 ***
UNRATE.l2 2.97363 9.43727 0.315 0.752804
DSPIC96.l2 0.01885 0.02000 0.943 0.346262
PCE.l2 0.19367 0.05529 3.503 0.000496 ***
GS10.l2 -6.49579 6.33156 -1.026 0.305350
FEDFUNDS.l2 -0.49691 3.73486 -0.133 0.894203
CPIAUCSL.l3 3.02337 3.05594 0.989 0.322909
UNRATE.l3 12.70421 6.69125 1.899 0.058111 .
DSPIC96.l3 -0.02594 0.01639 -1.582 0.114087
PCE.l3 -0.02060 0.04535 -0.454 0.649819
GS10.l3 8.28570 4.21260 1.967 0.049674 *
FEDFUNDS.l3 1.34115 2.32053 0.578 0.563524
const 21.37961 33.45567 0.639 0.523048
trend 0.62832 0.19889 3.159 0.001665 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 26.02 on 578 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 8.017e+05 on 19 and 578 DF, p-value: < 2.2e-16
Estimation results for equation GS10:
=====================================
GS10 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 7.595e-02 3.280e-02 2.316 0.02091 *
UNRATE.l1 -1.345e-01 6.848e-02 -1.965 0.04994 *
DSPIC96.l1 8.116e-05 1.692e-04 0.480 0.63165
PCE.l1 1.088e-03 4.590e-04 2.370 0.01811 *
GS10.l1 1.300e+00 4.295e-02 30.265 < 2e-16 ***
FEDFUNDS.l1 1.310e-02 2.528e-02 0.518 0.60442
CPIAUCSL.l2 -1.344e-01 5.076e-02 -2.647 0.00834 **
UNRATE.l2 2.030e-01 9.762e-02 2.079 0.03806 *
DSPIC96.l2 -7.918e-05 2.069e-04 -0.383 0.70201
PCE.l2 -8.839e-04 5.719e-04 -1.545 0.12281
GS10.l2 -5.622e-01 6.550e-02 -8.584 < 2e-16 ***
FEDFUNDS.l2 2.509e-02 3.863e-02 0.649 0.51628
CPIAUCSL.l3 5.247e-02 3.161e-02 1.660 0.09747 .
UNRATE.l3 -5.275e-02 6.922e-02 -0.762 0.44634
DSPIC96.l3 -1.253e-04 1.696e-04 -0.739 0.46036
PCE.l3 -1.757e-04 4.691e-04 -0.375 0.70805
GS10.l3 2.073e-01 4.358e-02 4.757 2.49e-06 ***
FEDFUNDS.l3 -9.962e-03 2.400e-02 -0.415 0.67830
const 7.471e-01 3.461e-01 2.159 0.03127 *
trend 3.641e-03 2.057e-03 1.770 0.07733 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2692 on 578 degrees of freedom
Multiple R-Squared: 0.9925, Adjusted R-squared: 0.9923
F-statistic: 4032 on 19 and 578 DF, p-value: < 2.2e-16
Estimation results for equation FEDFUNDS:
=========================================
FEDFUNDS = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -1.087e-02 5.599e-02 -0.194 0.84616
UNRATE.l1 -4.983e-01 1.169e-01 -4.262 2.37e-05 ***
DSPIC96.l1 -6.070e-05 2.889e-04 -0.210 0.83365
PCE.l1 1.475e-04 7.836e-04 0.188 0.85077
GS10.l1 5.137e-01 7.334e-02 7.005 6.88e-12 ***
FEDFUNDS.l1 1.318e+00 4.316e-02 30.538 < 2e-16 ***
CPIAUCSL.l2 -2.474e-02 8.666e-02 -0.285 0.77543
UNRATE.l2 5.383e-01 1.667e-01 3.230 0.00131 **
DSPIC96.l2 3.376e-04 3.532e-04 0.956 0.33946
PCE.l2 -9.879e-05 9.765e-04 -0.101 0.91945
GS10.l2 -5.653e-01 1.118e-01 -5.056 5.77e-07 ***
FEDFUNDS.l2 -4.970e-01 6.596e-02 -7.534 1.90e-13 ***
CPIAUCSL.l3 2.421e-02 5.397e-02 0.449 0.65386
UNRATE.l3 -7.198e-02 1.182e-01 -0.609 0.54271
DSPIC96.l3 -2.633e-04 2.895e-04 -0.909 0.36357
PCE.l3 -4.632e-05 8.009e-04 -0.058 0.95390
GS10.l3 1.205e-01 7.440e-02 1.619 0.10597
FEDFUNDS.l3 1.221e-01 4.098e-02 2.978 0.00302 **
const 4.190e-01 5.909e-01 0.709 0.47857
trend 3.908e-03 3.513e-03 1.113 0.26630
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4596 on 578 degrees of freedom
Multiple R-Squared: 0.9867, Adjusted R-squared: 0.9862
F-statistic: 2254 on 19 and 578 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 0.1319789 -0.0009514 -3.73603 3.2543 0.007906 0.005619
UNRATE -0.0009514 0.0262322 -0.03135 -0.4439 -0.006329 -0.011396
DSPIC96 -3.7360288 -0.0313497 4422.68797 13.7281 -0.167113 -0.932497
PCE 3.2543067 -0.4439276 13.72809 677.2438 0.300133 0.388906
GS10 0.0079055 -0.0063292 -0.16711 0.3001 0.072468 0.036450
FEDFUNDS 0.0056189 -0.0113963 -0.93250 0.3889 0.036450 0.211253
Correlation matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 1.00000 -0.016170 -0.154637 0.344218 0.080836 0.03365
UNRATE -0.01617 1.000000 -0.002911 -0.105323 -0.145164 -0.15309
DSPIC96 -0.15464 -0.002911 1.000000 0.007932 -0.009335 -0.03051
PCE 0.34422 -0.105323 0.007932 1.000000 0.042842 0.03251
GS10 0.08084 -0.145164 -0.009335 0.042842 1.000000 0.29459
FEDFUNDS 0.03365 -0.153089 -0.030507 0.032514 0.294595 1.00000K-Fold Cross Validation and Model Diagnostics
# Define the number of folds for cross-validation
k <- 5
# Define the p values to test
p_values <- c(1, 2, 3)
# Split the data into k folds
cv_folds <- cut(seq(1, nrow(var_ts1)), breaks = k, labels = FALSE)
# Initialize vectors to store RMSE and AIC values for each p value
rmse_vec <- numeric(length(p_values))
aic_vec <- numeric(length(p_values))
# Loop over p values and perform cross-validation
for (i in seq_along(p_values)) {
p <- p_values[i]
rmse_cv <- numeric(k)
aic_cv <- numeric(k)
for (j in 1:k) {
# Split the data into training and testing sets
train <- var_ts1[cv_folds != j, ]
test <- var_ts1[cv_folds == j, ]
# Fit the VAR model with the current p value
var_fit <- VAR(train, p = p)
# Make predictions for the testing set
pred <- predict(var_fit, n.ahead = nrow(test))$fcst
# Calculate RMSE and AIC for the current fold
rmse_cv[j] <- sqrt(mean((pred$CPIAUCSL - test[,1])^2))
aic_cv[j] <- AIC(var_fit)
}
# Calculate the mean RMSE and AIC across all folds for the current p value
rmse_vec[i] <- mean(rmse_cv)
aic_vec[i] <- mean(aic_cv)
}
# Create a table of RMSE and AIC values for each p value
results_table <- tibble(p_values, rmse_vec, aic_vec)
# Print the results table
kable(results_table, format = "markdown",
col.names = c("P Values", "Mean RMSE (5 Folds)", "Mean AIC (5 Folds)"), align = "c", digits = 2
)| P Values | Mean RMSE (5 Folds) | Mean AIC (5 Folds) |
|---|---|---|
| 1 | 151.16 | 11942.99 |
| 2 | 145.37 | 11658.50 |
| 3 | 144.87 | 11587.17 |
The VAR(3) model outputs the lowest Mean RMSE of 144.87 inflation from the 5-fold cross validation. However, it has the highest AIC score. Because test set performance is best and it is the simplest model, we shall choose the VAR(3) model as the best option.
Forecasting the chosen model (P=3)
final_var <- VAR(var_ts1, p = 3)
(fit.pr = predict(final_var, n.ahead = 5, ci = 0.95)) # 5 years ahead $CPIAUCSL
fcst lower upper CI
[1,] 259.3881 258.6743 260.1018 0.713728
[2,] 259.8701 258.5939 261.1463 1.276158
[3,] 260.3400 258.6425 262.0376 1.697582
[4,] 260.7761 258.7572 262.7949 2.018880
[5,] 261.2104 258.9234 263.4973 2.286950
$UNRATE
fcst lower upper CI
[1,] 3.482483 3.162932 3.802033 0.3195505
[2,] 3.462199 2.991376 3.933021 0.4708229
[3,] 3.476632 2.842868 4.110396 0.6337641
[4,] 3.489748 2.715837 4.263658 0.7739103
[5,] 3.513099 2.612477 4.413721 0.9006220
$DSPIC96
fcst lower upper CI
[1,] 15887.36 15756.60 16018.13 130.7634
[2,] 15906.22 15742.93 16069.51 163.2903
[3,] 15948.27 15766.61 16129.92 181.6532
[4,] 15986.92 15786.95 16186.88 199.9637
[5,] 16022.93 15806.49 16239.36 216.4349
$PCE
fcst lower upper CI
[1,] 14855.74 14804.34 14907.14 51.39997
[2,] 14908.94 14837.64 14980.25 71.30850
[3,] 14956.63 14865.87 15047.38 90.75522
[4,] 15003.92 14897.13 15110.71 106.78949
[5,] 15050.73 14929.64 15171.82 121.09257
$GS10
fcst lower upper CI
[1,] 1.715155 1.1865638 2.243747 0.5285914
[2,] 1.689191 0.7992903 2.579092 0.8899010
[3,] 1.667817 0.5549866 2.780648 1.1128308
[4,] 1.630803 0.3630515 2.898554 1.2677513
[5,] 1.588605 0.1911054 2.986104 1.3974992
$FEDFUNDS
fcst lower upper CI
[1,] 1.517328 0.61629815 2.418358 0.9010301
[2,] 1.552455 -0.05806836 3.162979 1.6105236
[3,] 1.591517 -0.53307984 3.716113 2.1245965
[4,] 1.593517 -0.90452339 4.091558 2.4980406
[5,] 1.579673 -1.20701980 4.366366 2.7866927fanchart(fit.pr) # plot prediction + error
The above plot showcases the forecasts for each variable present in the VAR(3) model, Yearly Inflation, Unemployment, Personal Consumption, Treasury Yield and Federal Funds Rate. The predicted forecast, from the years 2021 to 2025, for CPI is a good sign for the US due to the rising trend.
Let us visualize more closely the forecasts for the CPI from 2021 to 2025, corresponding to the VAR(3) model fitted on all years (1970-2020):
df_fvar_attack <- as.data.frame(fit.pr$fcst$CPIAUCSL)
# add year column
df_fvar_attack$Year <- c("2021", "2022", "2023", "2024", "2025")
(var_plot <- ggplot(data=df_fvar_attack, aes(x=Year, y=fcst, group = 1)) +
geom_line(aes(color="Forecast"), linewidth=1) +
geom_ribbon(aes(ymin=lower, ymax=upper, fill="Confidence Interval"), alpha=0.1) +
labs(title="VAR(3) Forecasts for CPI from 2021 to 2025",
y="CPI",
color="", fill="",
caption="Data Sources: FRED") +
scale_color_manual(values = c("Forecast"="red")) +
scale_fill_manual(values = c("95% Confidence Interval"="steelblue")) +
theme_minimal() +
theme(plot.caption.position = "plot"))
Building the VAR Model (Post COVID-19)
Time Series Plot
plot.ts(var_ts2 , main = "", xlab = "")
Pair Plots
# create scatterplot matrix using plotly
fig <- plot_ly(
data = as.data.frame(var_ts2),
type = "splom",
diagonal = list(visible = FALSE),
dimensions = list(
list(label = "CPI", values = ~CPIAUCSL),
list(label = "Unemployment", values = ~UNRATE),
list(label = "DisposableIncome", values = ~DSPIC96),
list(label = "PersonalConsumption", values = ~PCE),
list(label = "TreasuryYield", values = ~GS10),
list(label = "FederalFundsRate", values = ~FEDFUNDS)
)
) %>%
layout(hovermode = "x")
fig <- fig %>%
layout(
title = "Scatterplot Matrix of VAR Model Variables (Post COVID-19)",
xaxis = list(title = ""),
yaxis = list(title = "")
)
# display plot
figFitting the VAR Model
Here we use the VARselect() function to find the best p to fit VAR(p). We will choose a maximum lag of 10 and check which p value returns lowest AIC.
(var_result <- VARselect(var_ts2, lag.max = 10, type = "both"))$selection
AIC(n) HQ(n) SC(n) FPE(n)
10 4 2 10
$criteria
1 2 3 4 5 6
AIC(n) 12.10596 10.67409 10.54164 10.43213 10.43372 10.42279
HQ(n) 12.23718 10.90373 10.86970 10.85861 10.95863 11.04611
SC(n) 12.44385 11.26540 11.38636 11.53027 11.78529 12.02776
FPE(n) 180947.94882 43223.11299 37864.33401 33941.84237 34004.38817 33646.23719
7 8 9 10
AIC(n) 10.40910 10.41932 10.37218 10.33357
HQ(n) 11.13084 11.23949 11.29077 11.35058
SC(n) 12.26750 12.53113 12.73741 12.95222
FPE(n) 33204.88543 33567.30729 32047.59811 30864.68386Now, we will fit VAR(1), VAR(2), and VAR(3):
VAR(1) output:
summary(fit <- VAR(var_ts2, p=1, type="both"))
VAR Estimation Results:
=========================
Endogenous variables: CPIAUCSL, UNRATE, DSPIC96, PCE, GS10, FEDFUNDS
Deterministic variables: both
Sample size: 641
Log Likelihood: -9274.084
Roots of the characteristic polynomial:
0.999 0.9974 0.9974 0.9575 0.9152 0.6585
Call:
VAR(y = var_ts2, p = 1, type = "both")
Estimation results for equation CPIAUCSL:
=========================================
CPIAUCSL = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.004e+00 5.422e-03 185.098 <2e-16 ***
UNRATE.l1 -4.642e-03 1.376e-02 -0.337 0.7360
DSPIC96.l1 1.140e-04 6.431e-05 1.773 0.0768 .
PCE.l1 6.650e-05 3.760e-05 1.769 0.0775 .
GS10.l1 2.405e-02 2.320e-02 1.037 0.3002
FEDFUNDS.l1 2.948e-02 1.441e-02 2.045 0.0413 *
const -5.464e-01 3.362e-01 -1.625 0.1046
trend -4.218e-03 2.366e-03 -1.783 0.0751 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.49 on 633 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 1.952e+06 on 7 and 633 DF, p-value: < 2.2e-16
Estimation results for equation UNRATE:
=======================================
UNRATE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -7.237e-04 5.114e-03 -0.142 0.8875
UNRATE.l1 9.644e-01 1.298e-02 74.289 <2e-16 ***
DSPIC96.l1 -9.547e-05 6.066e-05 -1.574 0.1160
PCE.l1 3.030e-05 3.546e-05 0.854 0.3932
GS10.l1 -1.413e-02 2.188e-02 -0.646 0.5188
FEDFUNDS.l1 1.259e-02 1.360e-02 0.926 0.3547
const 6.086e-01 3.171e-01 1.919 0.0554 .
trend 1.350e-03 2.232e-03 0.605 0.5453
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4622 on 633 degrees of freedom
Multiple R-Squared: 0.9264, Adjusted R-squared: 0.9256
F-statistic: 1139 on 7 and 633 DF, p-value: < 2.2e-16
Estimation results for equation DSPIC96:
========================================
DSPIC96 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -12.55136 2.56069 -4.902 1.21e-06 ***
UNRATE.l1 4.17093 6.50028 0.642 0.521
DSPIC96.l1 0.68905 0.03037 22.687 < 2e-16 ***
PCE.l1 0.13983 0.01776 7.875 1.49e-14 ***
GS10.l1 -17.75539 10.95601 -1.621 0.106
FEDFUNDS.l1 10.10453 6.80730 1.484 0.138
const 1503.78315 158.78407 9.471 < 2e-16 ***
trend 7.70122 1.11749 6.892 1.34e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 231.4 on 633 degrees of freedom
Multiple R-Squared: 0.9965, Adjusted R-squared: 0.9964
F-statistic: 2.559e+04 on 7 and 633 DF, p-value: < 2.2e-16
Estimation results for equation PCE:
====================================
PCE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 2.903e+00 1.071e+00 2.710 0.006911 **
UNRATE.l1 9.673e+00 2.719e+00 3.558 0.000402 ***
DSPIC96.l1 7.236e-02 1.270e-02 5.696 1.88e-08 ***
PCE.l1 9.703e-01 7.427e-03 130.635 < 2e-16 ***
GS10.l1 -2.477e+00 4.583e+00 -0.541 0.589006
FEDFUNDS.l1 2.484e+00 2.847e+00 0.872 0.383275
const -3.949e+02 6.642e+01 -5.946 4.53e-09 ***
trend -1.752e+00 4.674e-01 -3.748 0.000195 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 96.8 on 633 degrees of freedom
Multiple R-Squared: 0.9996, Adjusted R-squared: 0.9996
F-statistic: 2.233e+05 on 7 and 633 DF, p-value: < 2.2e-16
Estimation results for equation GS10:
=====================================
GS10 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -2.657e-03 3.239e-03 -0.820 0.4123
UNRATE.l1 7.473e-03 8.222e-03 0.909 0.3637
DSPIC96.l1 -6.607e-05 3.841e-05 -1.720 0.0859 .
PCE.l1 3.731e-05 2.246e-05 1.661 0.0972 .
GS10.l1 9.671e-01 1.386e-02 69.788 <2e-16 ***
FEDFUNDS.l1 1.724e-02 8.610e-03 2.003 0.0456 *
const 4.340e-01 2.008e-01 2.161 0.0311 *
trend 1.281e-03 1.413e-03 0.907 0.3650
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2927 on 633 degrees of freedom
Multiple R-Squared: 0.9915, Adjusted R-squared: 0.9914
F-statistic: 1.05e+04 on 7 and 633 DF, p-value: < 2.2e-16
Estimation results for equation FEDFUNDS:
=========================================
FEDFUNDS = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -1.242e-02 5.813e-03 -2.137 0.03296 *
UNRATE.l1 -4.563e-02 1.476e-02 -3.092 0.00207 **
DSPIC96.l1 -3.499e-05 6.895e-05 -0.507 0.61201
PCE.l1 8.068e-05 4.031e-05 2.001 0.04577 *
GS10.l1 1.006e-01 2.487e-02 4.044 5.9e-05 ***
FEDFUNDS.l1 9.305e-01 1.545e-02 60.214 < 2e-16 ***
const 5.428e-01 3.604e-01 1.506 0.13259
trend 3.728e-03 2.537e-03 1.469 0.14219
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5253 on 633 degrees of freedom
Multiple R-Squared: 0.9823, Adjusted R-squared: 0.9821
F-statistic: 5030 on 7 and 633 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 0.24013 -0.05535 -15.686 16.695 0.03101 0.02665
UNRATE -0.05535 0.21361 36.255 -32.261 -0.01421 -0.03240
DSPIC96 -15.68635 36.25500 53555.024 -571.886 -1.06595 -3.35790
PCE 16.69544 -32.26071 -571.886 9369.730 3.31255 5.74350
GS10 0.03101 -0.01421 -1.066 3.313 0.08567 0.05487
FEDFUNDS 0.02665 -0.03240 -3.358 5.744 0.05487 0.27597
Correlation matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 1.0000 -0.2444 -0.13833 0.35198 0.21621 0.10354
UNRATE -0.2444 1.0000 0.33897 -0.72110 -0.10500 -0.13344
DSPIC96 -0.1383 0.3390 1.00000 -0.02553 -0.01574 -0.02762
PCE 0.3520 -0.7211 -0.02553 1.00000 0.11692 0.11295
GS10 0.2162 -0.1050 -0.01574 0.11692 1.00000 0.35682
FEDFUNDS 0.1035 -0.1334 -0.02762 0.11295 0.35682 1.00000VAR(2) output:
summary(fit <- VAR(var_ts2, p=2, type="both"))
VAR Estimation Results:
=========================
Endogenous variables: CPIAUCSL, UNRATE, DSPIC96, PCE, GS10, FEDFUNDS
Deterministic variables: both
Sample size: 640
Log Likelihood: -8765.803
Roots of the characteristic polynomial:
1.002 0.9938 0.976 0.9665 0.8687 0.8687 0.5569 0.4755 0.439 0.439 0.3769 0.3769
Call:
VAR(y = var_ts2, p = 2, type = "both")
Estimation results for equation CPIAUCSL:
=========================================
CPIAUCSL = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.4215516 0.0379593 37.449 < 2e-16 ***
UNRATE.l1 0.1853689 0.0597027 3.105 0.00199 **
DSPIC96.l1 -0.0000137 0.0000832 -0.165 0.86928
PCE.l1 0.0011975 0.0002791 4.290 2.07e-05 ***
GS10.l1 0.1185237 0.0627596 1.889 0.05942 .
FEDFUNDS.l1 0.0646079 0.0344637 1.875 0.06130 .
CPIAUCSL.l2 -0.4220773 0.0380908 -11.081 < 2e-16 ***
UNRATE.l2 -0.1829256 0.0605829 -3.019 0.00264 **
DSPIC96.l2 0.0001004 0.0000787 1.275 0.20264
PCE.l2 -0.0011885 0.0002694 -4.411 1.21e-05 ***
GS10.l2 -0.1108661 0.0627963 -1.765 0.07797 .
FEDFUNDS.l2 -0.0468105 0.0341387 -1.371 0.17081
const -0.3101875 0.3341981 -0.928 0.35368
trend -0.0012712 0.0022396 -0.568 0.57051
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4232 on 626 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 1.403e+06 on 13 and 626 DF, p-value: < 2.2e-16
Estimation results for equation UNRATE:
=======================================
UNRATE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 3.780e-02 3.322e-02 1.138 0.255642
UNRATE.l1 3.118e-01 5.225e-02 5.967 4.05e-09 ***
DSPIC96.l1 2.553e-04 7.281e-05 3.507 0.000486 ***
PCE.l1 -4.257e-03 2.443e-04 -17.428 < 2e-16 ***
GS10.l1 -1.138e-01 5.492e-02 -2.072 0.038711 *
FEDFUNDS.l1 -7.030e-02 3.016e-02 -2.331 0.020081 *
CPIAUCSL.l2 -2.805e-02 3.333e-02 -0.841 0.400461
UNRATE.l2 6.658e-01 5.302e-02 12.558 < 2e-16 ***
DSPIC96.l2 -8.537e-05 6.888e-05 -1.239 0.215657
PCE.l2 4.201e-03 2.358e-04 17.816 < 2e-16 ***
GS10.l2 9.876e-02 5.495e-02 1.797 0.072785 .
FEDFUNDS.l2 9.394e-02 2.988e-02 3.144 0.001744 **
const -7.496e-01 2.925e-01 -2.563 0.010604 *
trend -5.475e-03 1.960e-03 -2.793 0.005374 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3704 on 626 degrees of freedom
Multiple R-Squared: 0.9532, Adjusted R-squared: 0.9522
F-statistic: 980.7 on 13 and 626 DF, p-value: < 2.2e-16
Estimation results for equation DSPIC96:
========================================
DSPIC96 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -19.45821 16.87132 -1.153 0.2492
UNRATE.l1 -134.95087 26.53532 -5.086 4.85e-07 ***
DSPIC96.l1 0.44313 0.03698 11.984 < 2e-16 ***
PCE.l1 -1.13145 0.12407 -9.120 < 2e-16 ***
GS10.l1 3.50497 27.89400 0.126 0.9000
FEDFUNDS.l1 -7.43573 15.31766 -0.485 0.6275
CPIAUCSL.l2 16.36520 16.92976 0.967 0.3341
UNRATE.l2 134.87420 26.92654 5.009 7.13e-07 ***
DSPIC96.l2 0.48497 0.03498 13.864 < 2e-16 ***
PCE.l2 1.16933 0.11974 9.765 < 2e-16 ***
GS10.l2 -9.15256 27.91032 -0.328 0.7431
FEDFUNDS.l2 13.54166 15.17323 0.892 0.3725
const 368.11628 148.53700 2.478 0.0135 *
trend 1.88999 0.99539 1.899 0.0581 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 188.1 on 626 degrees of freedom
Multiple R-Squared: 0.9977, Adjusted R-squared: 0.9976
F-statistic: 2.082e+04 on 13 and 626 DF, p-value: < 2.2e-16
Estimation results for equation PCE:
====================================
PCE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 4.83697 7.72100 0.626 0.531236
UNRATE.l1 152.54985 12.14365 12.562 < 2e-16 ***
DSPIC96.l1 -0.02646 0.01692 -1.563 0.118449
PCE.l1 1.61059 0.05678 28.366 < 2e-16 ***
GS10.l1 21.97184 12.76543 1.721 0.085709 .
FEDFUNDS.l1 10.72636 7.00999 1.530 0.126483
CPIAUCSL.l2 -3.98752 7.74775 -0.515 0.606967
UNRATE.l2 -146.96614 12.32268 -11.926 < 2e-16 ***
DSPIC96.l2 0.05725 0.01601 3.576 0.000376 ***
PCE.l2 -0.62567 0.05480 -11.418 < 2e-16 ***
GS10.l2 -21.16124 12.77290 -1.657 0.098075 .
FEDFUNDS.l2 -12.19515 6.94389 -1.756 0.079536 .
const -167.34735 67.97660 -2.462 0.014091 *
trend -0.54154 0.45553 -1.189 0.234964
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 86.08 on 626 degrees of freedom
Multiple R-Squared: 0.9997, Adjusted R-squared: 0.9997
F-statistic: 1.517e+05 on 13 and 626 DF, p-value: < 2.2e-16
Estimation results for equation GS10:
=====================================
GS10 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 6.550e-02 2.465e-02 2.657 0.00809 **
UNRATE.l1 7.839e-03 3.877e-02 0.202 0.83985
DSPIC96.l1 -2.594e-05 5.403e-05 -0.480 0.63134
PCE.l1 4.549e-05 1.813e-04 0.251 0.80194
GS10.l1 1.270e+00 4.076e-02 31.161 < 2e-16 ***
FEDFUNDS.l1 -8.618e-03 2.238e-02 -0.385 0.70032
CPIAUCSL.l2 -6.637e-02 2.474e-02 -2.683 0.00749 **
UNRATE.l2 6.638e-03 3.934e-02 0.169 0.86607
DSPIC96.l2 -5.303e-05 5.111e-05 -1.038 0.29989
PCE.l2 -2.424e-05 1.750e-04 -0.139 0.88985
GS10.l2 -3.264e-01 4.078e-02 -8.004 5.85e-15 ***
FEDFUNDS.l2 3.381e-02 2.217e-02 1.525 0.12780
const 4.882e-01 2.170e-01 2.249 0.02484 *
trend 1.047e-03 1.454e-03 0.720 0.47177
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2748 on 626 degrees of freedom
Multiple R-Squared: 0.9926, Adjusted R-squared: 0.9924
F-statistic: 6417 on 13 and 626 DF, p-value: < 2.2e-16
Estimation results for equation FEDFUNDS:
=========================================
FEDFUNDS = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -2.041e-02 4.121e-02 -0.495 0.6207
UNRATE.l1 -1.106e-01 6.482e-02 -1.706 0.0884 .
DSPIC96.l1 6.588e-05 9.033e-05 0.729 0.4661
PCE.l1 -3.047e-04 3.031e-04 -1.005 0.3152
GS10.l1 5.524e-01 6.814e-02 8.107 2.74e-15 ***
FEDFUNDS.l1 1.266e+00 3.742e-02 33.828 < 2e-16 ***
CPIAUCSL.l2 1.813e-02 4.136e-02 0.438 0.6613
UNRATE.l2 7.872e-02 6.578e-02 1.197 0.2319
DSPIC96.l2 -2.405e-05 8.545e-05 -0.281 0.7785
PCE.l2 3.212e-04 2.925e-04 1.098 0.2726
GS10.l2 -4.702e-01 6.818e-02 -6.897 1.30e-11 ***
FEDFUNDS.l2 -3.376e-01 3.707e-02 -9.108 < 2e-16 ***
const 5.425e-02 3.629e-01 0.150 0.8812
trend -3.988e-04 2.432e-03 -0.164 0.8698
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4595 on 626 degrees of freedom
Multiple R-Squared: 0.9866, Adjusted R-squared: 0.9863
F-statistic: 3550 on 13 and 626 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 0.17910 -0.016725 -7.566 10.897 0.016536 0.01143
UNRATE -0.01672 0.137164 13.924 -21.288 -0.006801 -0.02031
DSPIC96 -7.56559 13.924086 35380.471 1741.521 1.640781 -1.27537
PCE 10.89689 -21.288001 1741.521 7409.914 2.247045 4.56767
GS10 0.01654 -0.006801 1.641 2.247 0.075540 0.03990
FEDFUNDS 0.01143 -0.020314 -1.275 4.568 0.039898 0.21114
Correlation matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 1.00000 -0.10671 -0.09504 0.29912 0.14217 0.05877
UNRATE -0.10671 1.00000 0.19988 -0.66774 -0.06682 -0.11937
DSPIC96 -0.09504 0.19988 1.00000 0.10756 0.03174 -0.01476
PCE 0.29912 -0.66774 0.10756 1.00000 0.09498 0.11548
GS10 0.14217 -0.06682 0.03174 0.09498 1.00000 0.31592
FEDFUNDS 0.05877 -0.11937 -0.01476 0.11548 0.31592 1.00000VAR(3) output:
summary(fit <- VAR(var_ts2, p=3, type="both"))
VAR Estimation Results:
=========================
Endogenous variables: CPIAUCSL, UNRATE, DSPIC96, PCE, GS10, FEDFUNDS
Deterministic variables: both
Sample size: 639
Log Likelihood: -8674.334
Roots of the characteristic polynomial:
1.002 0.9966 0.9822 0.9673 0.8943 0.8943 0.5423 0.5423 0.5056 0.5056 0.4748 0.4748 0.4181 0.4181 0.244 0.244 0.07907 0.06819
Call:
VAR(y = var_ts2, p = 3, type = "both")
Estimation results for equation CPIAUCSL:
=========================================
CPIAUCSL = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.455e+00 4.345e-02 33.483 < 2e-16 ***
UNRATE.l1 2.048e-01 6.736e-02 3.041 0.00246 **
DSPIC96.l1 -3.795e-05 9.929e-05 -0.382 0.70245
PCE.l1 1.214e-03 2.990e-04 4.059 5.56e-05 ***
GS10.l1 1.449e-01 6.554e-02 2.211 0.02743 *
FEDFUNDS.l1 5.266e-02 3.901e-02 1.350 0.17754
CPIAUCSL.l2 -5.571e-01 7.115e-02 -7.830 2.13e-14 ***
UNRATE.l2 -2.003e-01 8.964e-02 -2.234 0.02581 *
DSPIC96.l2 6.915e-05 9.327e-05 0.741 0.45877
PCE.l2 -1.008e-03 4.604e-04 -2.189 0.02900 *
GS10.l2 -1.549e-01 1.006e-01 -1.540 0.12406
FEDFUNDS.l2 -4.306e-03 6.025e-02 -0.071 0.94304
CPIAUCSL.l3 1.032e-01 4.211e-02 2.451 0.01452 *
UNRATE.l3 -5.081e-03 6.993e-02 -0.073 0.94210
DSPIC96.l3 6.730e-05 9.161e-05 0.735 0.46285
PCE.l3 -1.957e-04 3.403e-04 -0.575 0.56531
GS10.l3 2.370e-02 6.698e-02 0.354 0.72355
FEDFUNDS.l3 -3.266e-02 3.706e-02 -0.882 0.37838
const -3.892e-01 3.517e-01 -1.107 0.26884
trend -2.102e-03 2.339e-03 -0.899 0.36922
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.423 on 619 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 9.571e+05 on 19 and 619 DF, p-value: < 2.2e-16
Estimation results for equation UNRATE:
=======================================
UNRATE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 7.876e-02 3.753e-02 2.099 0.03624 *
UNRATE.l1 2.777e-01 5.818e-02 4.774 2.26e-06 ***
DSPIC96.l1 3.058e-04 8.575e-05 3.566 0.00039 ***
PCE.l1 -4.667e-03 2.583e-04 -18.068 < 2e-16 ***
GS10.l1 -1.161e-01 5.660e-02 -2.051 0.04072 *
FEDFUNDS.l1 -4.357e-02 3.370e-02 -1.293 0.19643
CPIAUCSL.l2 -1.153e-01 6.145e-02 -1.876 0.06113 .
UNRATE.l2 9.307e-01 7.742e-02 12.022 < 2e-16 ***
DSPIC96.l2 -2.363e-04 8.056e-05 -2.933 0.00348 **
PCE.l2 5.164e-03 3.976e-04 12.987 < 2e-16 ***
GS10.l2 1.165e-01 8.685e-02 1.342 0.18024
FEDFUNDS.l2 3.070e-02 5.204e-02 0.590 0.55544
CPIAUCSL.l3 4.385e-02 3.637e-02 1.206 0.22838
UNRATE.l3 -2.365e-01 6.039e-02 -3.916 0.00010 ***
DSPIC96.l3 5.552e-05 7.912e-05 0.702 0.48314
PCE.l3 -5.290e-04 2.939e-04 -1.800 0.07235 .
GS10.l3 -9.171e-03 5.785e-02 -0.159 0.87408
FEDFUNDS.l3 3.308e-02 3.200e-02 1.034 0.30166
const -5.142e-01 3.037e-01 -1.693 0.09093 .
trend -4.222e-03 2.020e-03 -2.090 0.03705 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3653 on 619 degrees of freedom
Multiple R-Squared: 0.9549, Adjusted R-squared: 0.9535
F-statistic: 689.6 on 19 and 619 DF, p-value: < 2.2e-16
Estimation results for equation DSPIC96:
========================================
DSPIC96 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 -2.84105 19.15148 -0.148 0.882118
UNRATE.l1 -175.64362 29.69048 -5.916 5.47e-09 ***
DSPIC96.l1 0.38830 0.04376 8.873 < 2e-16 ***
PCE.l1 -1.19554 0.13181 -9.070 < 2e-16 ***
GS10.l1 11.43483 28.88638 0.396 0.692348
FEDFUNDS.l1 -12.30414 17.19537 -0.716 0.474539
CPIAUCSL.l2 -15.35811 31.35946 -0.490 0.624489
UNRATE.l2 134.52057 39.50940 3.405 0.000705 ***
DSPIC96.l2 0.49485 0.04111 12.037 < 2e-16 ***
PCE.l2 0.75594 0.20292 3.725 0.000213 ***
GS10.l2 -16.45059 44.32124 -0.371 0.710640
FEDFUNDS.l2 17.17192 26.55585 0.647 0.518108
CPIAUCSL.l3 16.39397 18.56128 0.883 0.377452
UNRATE.l3 39.16077 30.82043 1.271 0.204344
DSPIC96.l3 0.07422 0.04038 1.838 0.066492 .
PCE.l3 0.46848 0.14997 3.124 0.001869 **
GS10.l3 2.12261 29.52092 0.072 0.942703
FEDFUNDS.l3 0.61079 16.33225 0.037 0.970180
const 229.41415 154.99244 1.480 0.139338
trend 1.08535 1.03104 1.053 0.292900
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 186.4 on 619 degrees of freedom
Multiple R-Squared: 0.9978, Adjusted R-squared: 0.9977
F-statistic: 1.445e+04 on 19 and 619 DF, p-value: < 2.2e-16
Estimation results for equation PCE:
====================================
PCE = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 10.31227 8.80046 1.172 0.241734
UNRATE.l1 134.95588 13.64333 9.892 < 2e-16 ***
DSPIC96.l1 -0.04224 0.02011 -2.101 0.036073 *
PCE.l1 1.57997 0.06057 26.086 < 2e-16 ***
GS10.l1 25.32738 13.27383 1.908 0.056845 .
FEDFUNDS.l1 8.18212 7.90159 1.036 0.300839
CPIAUCSL.l2 -10.24779 14.41026 -0.711 0.477262
UNRATE.l2 -143.31706 18.15531 -7.894 1.34e-14 ***
DSPIC96.l2 0.06398 0.01889 3.387 0.000751 ***
PCE.l2 -0.78580 0.09324 -8.427 2.48e-16 ***
GS10.l2 -29.36288 20.36644 -1.442 0.149885
FEDFUNDS.l2 -6.73688 12.20291 -0.552 0.581099
CPIAUCSL.l3 1.17986 8.52926 0.138 0.890024
UNRATE.l3 13.17155 14.16257 0.930 0.352720
DSPIC96.l3 0.01923 0.01855 1.036 0.300392
PCE.l3 0.18779 0.06892 2.725 0.006613 **
GS10.l3 6.21115 13.56542 0.458 0.647209
FEDFUNDS.l3 -3.55427 7.50498 -0.474 0.635961
const -213.76325 71.22194 -3.001 0.002796 **
trend -0.80677 0.47378 -1.703 0.089100 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 85.67 on 619 degrees of freedom
Multiple R-Squared: 0.9997, Adjusted R-squared: 0.9997
F-statistic: 1.045e+05 on 19 and 619 DF, p-value: < 2.2e-16
Estimation results for equation GS10:
=====================================
GS10 = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 1.057e-01 2.760e-02 3.831 0.000141 ***
UNRATE.l1 -3.024e-02 4.278e-02 -0.707 0.479956
DSPIC96.l1 3.271e-05 6.306e-05 0.519 0.604131
PCE.l1 -6.582e-05 1.899e-04 -0.347 0.729053
GS10.l1 1.318e+00 4.162e-02 31.658 < 2e-16 ***
FEDFUNDS.l1 1.692e-02 2.478e-02 0.683 0.494916
CPIAUCSL.l2 -1.467e-01 4.519e-02 -3.246 0.001232 **
UNRATE.l2 2.460e-02 5.693e-02 0.432 0.665769
DSPIC96.l2 -2.008e-05 5.924e-05 -0.339 0.734755
PCE.l2 1.131e-05 2.924e-04 0.039 0.969154
GS10.l2 -5.874e-01 6.386e-02 -9.197 < 2e-16 ***
FEDFUNDS.l2 1.828e-02 3.827e-02 0.478 0.632935
CPIAUCSL.l3 3.899e-02 2.675e-02 1.458 0.145383
UNRATE.l3 1.759e-02 4.441e-02 0.396 0.692251
DSPIC96.l3 -9.884e-05 5.818e-05 -1.699 0.089843 .
PCE.l3 8.885e-05 2.161e-04 0.411 0.681090
GS10.l3 2.230e-01 4.254e-02 5.242 2.18e-07 ***
FEDFUNDS.l3 -1.294e-02 2.353e-02 -0.550 0.582740
const 5.077e-01 2.233e-01 2.273 0.023345 *
trend 1.378e-03 1.486e-03 0.928 0.353938
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2686 on 619 degrees of freedom
Multiple R-Squared: 0.993, Adjusted R-squared: 0.9927
F-statistic: 4597 on 19 and 619 DF, p-value: < 2.2e-16
Estimation results for equation FEDFUNDS:
=========================================
FEDFUNDS = CPIAUCSL.l1 + UNRATE.l1 + DSPIC96.l1 + PCE.l1 + GS10.l1 + FEDFUNDS.l1 + CPIAUCSL.l2 + UNRATE.l2 + DSPIC96.l2 + PCE.l2 + GS10.l2 + FEDFUNDS.l2 + CPIAUCSL.l3 + UNRATE.l3 + DSPIC96.l3 + PCE.l3 + GS10.l3 + FEDFUNDS.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
CPIAUCSL.l1 2.062e-02 4.670e-02 0.442 0.65889
UNRATE.l1 -1.674e-01 7.239e-02 -2.312 0.02110 *
DSPIC96.l1 1.188e-04 1.067e-04 1.114 0.26582
PCE.l1 -5.239e-04 3.214e-04 -1.630 0.10360
GS10.l1 5.419e-01 7.043e-02 7.693 5.68e-14 ***
FEDFUNDS.l1 1.339e+00 4.193e-02 31.941 < 2e-16 ***
CPIAUCSL.l2 -4.525e-02 7.646e-02 -0.592 0.55423
UNRATE.l2 1.624e-01 9.633e-02 1.686 0.09225 .
DSPIC96.l2 -3.545e-05 1.002e-04 -0.354 0.72373
PCE.l2 4.968e-04 4.948e-04 1.004 0.31575
GS10.l2 -5.942e-01 1.081e-01 -5.499 5.60e-08 ***
FEDFUNDS.l2 -5.150e-01 6.475e-02 -7.953 8.68e-15 ***
CPIAUCSL.l3 2.110e-02 4.526e-02 0.466 0.64114
UNRATE.l3 -2.912e-02 7.515e-02 -0.388 0.69850
DSPIC96.l3 -6.282e-05 9.845e-05 -0.638 0.52365
PCE.l3 6.049e-05 3.657e-04 0.165 0.86866
GS10.l3 1.297e-01 7.198e-02 1.802 0.07208 .
FEDFUNDS.l3 1.110e-01 3.982e-02 2.786 0.00549 **
const 1.694e-01 3.779e-01 0.448 0.65413
trend 1.320e-04 2.514e-03 0.053 0.95814
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4546 on 619 degrees of freedom
Multiple R-Squared: 0.987, Adjusted R-squared: 0.9866
F-statistic: 2481 on 19 and 619 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 0.17893 -0.016960 -7.857 11.013 0.015729 0.01174
UNRATE -0.01696 0.133468 13.591 -21.714 -0.007292 -0.02283
DSPIC96 -7.85739 13.590859 34759.020 1384.812 1.438151 -1.60584
PCE 11.01265 -21.714480 1384.812 7339.628 2.122663 4.45476
GS10 0.01573 -0.007292 1.438 2.123 0.072170 0.03731
FEDFUNDS 0.01174 -0.022825 -1.606 4.455 0.037308 0.20665
Correlation matrix of residuals:
CPIAUCSL UNRATE DSPIC96 PCE GS10 FEDFUNDS
CPIAUCSL 1.00000 -0.1098 -0.09963 0.30389 0.13841 0.06107
UNRATE -0.10975 1.0000 0.19954 -0.69378 -0.07430 -0.13744
DSPIC96 -0.09963 0.1995 1.00000 0.08670 0.02871 -0.01895
PCE 0.30389 -0.6938 0.08670 1.00000 0.09223 0.11439
GS10 0.13841 -0.0743 0.02871 0.09223 1.00000 0.30550
FEDFUNDS 0.06107 -0.1374 -0.01895 0.11439 0.30550 1.00000K-Fold Cross Validation and Model Diagnostics
# Define the number of folds for cross-validation
k <- 5
# Define the p values to test
p_values <- c(1, 2, 3)
# Split the data into k folds
cv_folds <- cut(seq(1, nrow(var_ts2)), breaks = k, labels = FALSE)
# Initialize vectors to store RMSE and AIC values for each p value
rmse_vec <- numeric(length(p_values))
aic_vec <- numeric(length(p_values))
# Loop over p values and perform cross-validation
for (i in seq_along(p_values)) {
p <- p_values[i]
rmse_cv <- numeric(k)
aic_cv <- numeric(k)
for (j in 1:k) {
# Split the data into training and testing sets
train <- var_ts2[cv_folds != j, ]
test <- var_ts2[cv_folds == j, ]
# Fit the VAR model with the current p value
var_fit <- VAR(train, p = p)
# Make predictions for the testing set
pred <- predict(var_fit, n.ahead = nrow(test))$fcst
# Calculate RMSE and AIC for the current fold
rmse_cv[j] <- sqrt(mean((pred$CPIAUCSL - test[,1])^2))
aic_cv[j] <- AIC(var_fit)
}
# Calculate the mean RMSE and AIC across all folds for the current p value
rmse_vec[i] <- mean(rmse_cv)
aic_vec[i] <- mean(aic_cv)
}
# Create a table of RMSE and AIC values for each p value
results_table <- tibble(p_values, rmse_vec, aic_vec)
# Print the results table
kable(results_table, format = "markdown",
col.names = c("P Values", "Mean RMSE (5 Folds)", "Mean AIC (5 Folds)"), align = "c", digits = 2
)| P Values | Mean RMSE (5 Folds) | Mean AIC (5 Folds) |
|---|---|---|
| 1 | 214.07 | 15472.03 |
| 2 | 197.77 | 14821.86 |
| 3 | 202.50 | 14714.38 |
The VAR(2) model outputs the lowest Mean RMSE of 197.84 from the 5-fold cross validation. However, it has the highest AIC score. Because predictive performance is best and it is the simplest model, we shall choose the VAR(2) model as the best option.
Forecasting the chosen model (P=2)
final_var <- VAR(var_ts2, p = 2)
(fit.pr = predict(final_var, n.ahead = 5, ci = 0.95)) # 5 years ahead $CPIAUCSL
fcst lower upper CI
[1,] 304.5980 303.7690 305.4270 0.8290186
[2,] 305.3897 303.8877 306.8916 1.5019366
[3,] 306.2069 304.1124 308.3014 2.0945028
[4,] 307.0482 304.4439 309.6525 2.6043236
[5,] 307.9016 304.8500 310.9531 3.0515667
$UNRATE
fcst lower upper CI
[1,] 3.668710 2.938897 4.398523 0.729813
[2,] 3.896853 2.746868 5.046838 1.149985
[3,] 4.091349 2.702882 5.479815 1.388466
[4,] 4.241375 2.694821 5.787928 1.546553
[5,] 4.385280 2.700831 6.069729 1.684449
$DSPIC96
fcst lower upper CI
[1,] 16925.05 16555.62 17294.47 369.4284
[2,] 17011.23 16589.17 17433.30 422.0644
[3,] 17104.42 16609.96 17598.88 494.4611
[4,] 17175.07 16644.95 17705.19 530.1203
[5,] 17252.02 16681.60 17822.45 570.4264
$PCE
fcst lower upper CI
[1,] 18508.30 18339.53 18677.07 168.7710
[2,] 18515.66 18239.26 18792.06 276.4016
[3,] 18540.34 18202.74 18877.95 337.6041
[4,] 18572.68 18194.11 18951.25 378.5740
[5,] 18606.08 18190.87 19021.29 415.2084
$GS10
fcst lower upper CI
[1,] 3.806431 3.267952 4.344910 0.5384791
[2,] 3.832180 2.955379 4.708980 0.8768005
[3,] 3.844942 2.710557 4.979328 1.1343854
[4,] 3.852638 2.512646 5.192629 1.3399920
[5,] 3.856427 2.344649 5.368205 1.5117781
$FEDFUNDS
fcst lower upper CI
[1,] 5.076313 4.176405 5.976221 0.8999079
[2,] 5.015310 3.460506 6.570113 1.5548037
[3,] 4.911707 2.823874 6.999540 2.0878330
[4,] 4.787958 2.272402 7.303515 2.5155566
[5,] 4.660179 1.798935 7.521422 2.8612435fanchart(fit.pr) # plot prediction + error
The above plot showcases the forecasts for each variable present in the VAR(2) model, Yearly Inflation, Unemployment, Personal Consumption, Treasury Yield and Federal Funds Rate. The above plot showcases the forecasts for each variable present in the VAR(3) model, Yearly Inflation, Unemployment, Personal Consumption, Treasury Yield and Federal Funds Rate. The predicted forecast, from the years 2021 to 2025, for CPI is a good sign for the US due to the rising trend.
Let us visualize more closely the forecasts for the CPI from 2021 to 2025, corresponding to the VAR(2) model fitted on Pre COVID-19 and post-COVID-19 years (2001-2020):
df_fvar_attack <- as.data.frame(fit.pr$fcst$CPIAUCSL)
# add year column
df_fvar_attack$Year <- c("2021", "2022", "2023", "2024", "2025")
(var_plot <- ggplot(data=df_fvar_attack, aes(x=Year, y=fcst, group = 1)) +
geom_line(aes(color="Forecast"), linewidth=1) +
geom_ribbon(aes(ymin=lower, ymax=upper, fill="Confidence Interval"), alpha=0.1) +
labs(title="VAR(3) Forecasts for CPI from 2021 to 2025",
y="CPI",
color="", fill="",
caption="Data Sources: FRED") +
scale_color_manual(values = c("Forecast"="red")) +
scale_fill_manual(values = c("95% Confidence Interval"="steelblue")) +
theme_minimal() +
theme(plot.caption.position = "plot"))
Manual ARIMAX Modelling (1970-2020: Pre COVID)
The ARIMAX model being analyzed in this section is:
CPI ~ Unemployment Rate + Disposable Income + Personal Consumption + Treasury Yield + Federal Funds Rate
Regression Summary and Fitting ARIMA to Residuals
fit.reg <- lm(CPIAUCSL ~ ., data =var_ts1)
summary(fit.reg)
Call:
lm(formula = CPIAUCSL ~ ., data = var_ts1)
Residuals:
Min 1Q Median 3Q Max
-22.2129 -6.5241 -0.9476 5.3942 22.4885
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.175e+02 5.669e+00 -20.732 < 2e-16 ***
UNRATE 1.303e+00 2.764e-01 4.713 3.05e-06 ***
DSPIC96 3.523e-02 1.272e-03 27.691 < 2e-16 ***
PCE -1.216e-02 1.071e-03 -11.350 < 2e-16 ***
GS10 3.938e+00 3.860e-01 10.200 < 2e-16 ***
FEDFUNDS -1.170e+00 2.574e-01 -4.545 6.65e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 8.533 on 595 degrees of freedom
Multiple R-squared: 0.9835, Adjusted R-squared: 0.9834
F-statistic: 7096 on 5 and 595 DF, p-value: < 2.2e-16res.fit<-ts(residuals(fit.reg),star=decimal_date(as.Date("1970-01-01",format = "%Y-%m-%d")),frequency = 1)
# Then look at the residuals
res.fit %>% ggtsdisplay() # no need to difference
#q=1,3 Q=1 , p=1,2, P=1,2
#write a funtion
SARIMA.c=function(p1,p2,q1,q2,P1,P2,Q1,Q2,d1,d2,data){
temp=c()
d=0
D=0
s=12
i=1
temp= data.frame()
ls=matrix(rep(NA,9*44),nrow=44)
for (p in p1:p2)
{
for(q in q1:q2)
{
for(P in P1:P2)
{
for(Q in Q1:Q2)
{
for(d in d1:d2)
{
if(p+d+q+P+D+Q<=8)
{
model<- Arima(data,order=c(p-1,d,q-1),seasonal=c(P-1,D,Q-1))
ls[i,]= c(p-1,d,q-1,P-1,D,Q-1,model$aic,model$bic,model$aicc)
i=i+1
#print(i)
}
}
}
}
}
}
temp= as.data.frame(ls)
names(temp)= c("p","d","q","P","D","Q","AIC","BIC","AICc")
temp
}
##q=1,3 Q=0 p=1,2 P=0 d=0
output=SARIMA.c(p1=1,p2=3,q1=1,q2=4,P1=1,P2=3,Q1=1,Q2=2,d1=0,d2=0,data=res.fit)
output p d q P D Q AIC BIC AICc
1 0 0 0 0 0 0 4280.507 4289.305 4280.527
2 0 0 0 0 0 1 4280.507 4289.305 4280.527
3 0 0 0 1 0 0 4280.507 4289.305 4280.527
4 0 0 0 1 0 1 4280.507 4289.305 4280.527
5 0 0 0 2 0 0 4280.507 4289.305 4280.527
6 0 0 0 2 0 1 4280.507 4289.305 4280.527
7 0 0 1 0 0 0 3716.284 3729.480 3716.324
8 0 0 1 0 0 1 3716.284 3729.480 3716.324
9 0 0 1 1 0 0 3716.284 3729.480 3716.324
10 0 0 1 1 0 1 3716.284 3729.480 3716.324
11 0 0 1 2 0 0 3716.284 3729.480 3716.324
12 0 0 1 2 0 1 3716.284 3729.480 3716.324
13 0 0 2 0 0 0 3435.150 3452.744 3435.217
14 0 0 2 0 0 1 3435.150 3452.744 3435.217
15 0 0 2 1 0 0 3435.150 3452.744 3435.217
16 0 0 2 1 0 1 3435.150 3452.744 3435.217
17 0 0 2 2 0 0 3435.150 3452.744 3435.217
18 0 0 3 0 0 0 3276.820 3298.813 3276.921
19 0 0 3 0 0 1 3276.820 3298.813 3276.921
20 0 0 3 1 0 0 3276.820 3298.813 3276.921
21 1 0 0 0 0 0 2942.581 2955.776 2942.621
22 1 0 0 0 0 1 2942.581 2955.776 2942.621
23 1 0 0 1 0 0 2942.581 2955.776 2942.621
24 1 0 0 1 0 1 2942.581 2955.776 2942.621
25 1 0 0 2 0 0 2942.581 2955.776 2942.621
26 1 0 0 2 0 1 2942.581 2955.776 2942.621
27 1 0 1 0 0 0 2939.053 2956.648 2939.121
28 1 0 1 0 0 1 2939.053 2956.648 2939.121
29 1 0 1 1 0 0 2939.053 2956.648 2939.121
30 1 0 1 1 0 1 2939.053 2956.648 2939.121
31 1 0 1 2 0 0 2939.053 2956.648 2939.121
32 1 0 2 0 0 0 2924.167 2946.160 2924.268
33 1 0 2 0 0 1 2924.167 2946.160 2924.268
34 1 0 2 1 0 0 2924.167 2946.160 2924.268
35 1 0 3 0 0 0 2923.576 2949.968 2923.718
36 2 0 0 0 0 0 2940.801 2958.395 2940.868
37 2 0 0 0 0 1 2940.801 2958.395 2940.868
38 2 0 0 1 0 0 2940.801 2958.395 2940.868
39 2 0 0 1 0 1 2940.801 2958.395 2940.868
40 2 0 0 2 0 0 2940.801 2958.395 2940.868
41 2 0 1 0 0 0 2926.113 2948.106 2926.214
42 2 0 1 0 0 1 2926.113 2948.106 2926.214
43 2 0 1 1 0 0 2926.113 2948.106 2926.214
44 2 0 2 0 0 0 2923.837 2950.229 2923.979ARIMA(1,0,3)
output[which.min(output$AIC),] p d q P D Q AIC BIC AICc
35 1 0 3 0 0 0 2923.576 2949.968 2923.718output[which.min(output$BIC),] p d q P D Q AIC BIC AICc
32 1 0 2 0 0 0 2924.167 2946.16 2924.268output[which.min(output$AICc),] p d q P D Q AIC BIC AICc
35 1 0 3 0 0 0 2923.576 2949.968 2923.718Model Diagnostics
model_outputar2 <- capture.output(sarima(res.fit, 1,0,3, 0,0,0))
cat(model_outputar2[57:89], model_outputar2[length(model_outputar2)], sep = "\n")iter 27 value 1.003339
iter 28 value 1.003338
iter 29 value 1.003338
iter 30 value 1.003338
iter 30 value 1.003338
iter 30 value 1.003338
final value 1.003338
converged
$fit
Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
xreg = xmean, include.mean = FALSE, transform.pars = trans, fixed = fixed,
optim.control = list(trace = trc, REPORT = 1, reltol = tol))
Coefficients:
ar1 ma1 ma2 ma3 xmean
0.9784 -0.1445 -0.1657 -0.0644 -0.557
s.e. 0.0091 0.0418 0.0428 0.0400 3.008
sigma^2 estimated as 7.409: log likelihood = -1455.79, aic = 2923.58
$degrees_of_freedom
[1] 596
$ttable
Estimate SE t.value p.value
ar1 0.9784 0.0091 107.3409 0.0000
ma1 -0.1445 0.0418 -3.4592 0.0006
ma2 -0.1657 0.0428 -3.8684 0.0001
ma3 -0.0644 0.0400 -1.6100 0.1079
xmean -0.5570 3.0080 -0.1852 0.8532arimaModel_1 <- arima(res.fit, order = c(1,0,3))
forecast1=predict(arimaModel_1, 5)
# create df with fcast preds and +-1.96 SE for 95% CI Bands
farimax_df <- data.frame(
Year = 2021:2025,
fcst = as.numeric(forecast1$pred),
lower = as.numeric(forecast1$pred - 1.96 * forecast1$se),
upper = as.numeric(forecast1$pred + 1.96 * forecast1$se)
)
(arimax_plot <- ggplot(data=farimax_df, aes(x=Year, y=fcst, group = 1)) +
geom_line(aes(color="Forecast"), linewidth=1) +
geom_ribbon(aes(ymin=lower, ymax=upper, fill="Confidence Interval"), alpha=0.1) +
labs(title="ARIMA(1,0,3) Forecasts for CPI from 2021 to 2025",
y="CPI",
color="", fill="",
caption="Data Sources: FRED") +
scale_color_manual(values = c("Forecast"="red")) +
scale_fill_manual(values = c("95% Confidence Interval"="steelblue")) +
theme_minimal() +
theme(plot.caption.position = "plot", plot.caption = element_text(size=8)))
Manual ARIMAX Modelling (2020-2023: Post COVID-19 )
The ARIMAX model being analyzed in this section is:
CPI ~ Unemployment Rate + Disposable Income + Personal Consumption + Treasury Yield + Federal Funds Rate
Regression Summary and Fitting ARIMA to Residuals
fit.reg <- lm(CPIAUCSL ~ ., data =var_ts2)
summary(fit.reg)
Call:
lm(formula = CPIAUCSL ~ ., data = var_ts2)
Residuals:
Min 1Q Median 3Q Max
-92.110 -8.310 1.465 8.365 23.493
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.943e+01 4.893e+00 -8.057 3.86e-15 ***
UNRATE -1.008e+00 3.170e-01 -3.179 0.00155 **
DSPIC96 1.763e-02 8.892e-04 19.828 < 2e-16 ***
PCE 2.064e-03 7.002e-04 2.948 0.00331 **
GS10 6.466e+00 4.687e-01 13.797 < 2e-16 ***
FEDFUNDS -2.986e+00 3.140e-01 -9.510 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 11.43 on 636 degrees of freedom
Multiple R-squared: 0.9748, Adjusted R-squared: 0.9746
F-statistic: 4919 on 5 and 636 DF, p-value: < 2.2e-16res.fit<-ts(residuals(fit.reg),star=decimal_date(as.Date("1970-01-01",format = "%Y-%m-%d")),frequency = 1)
# Then look at the residuals
res.fit %>% ggtsdisplay() # no need to difference
#q=1,3 Q=1 , p=1,2, P=1,2
#write a funtion
SARIMA.c=function(p1,p2,q1,q2,P1,P2,Q1,Q2,d1,d2,data){
temp=c()
d=0
D=0
s=12
i=1
temp= data.frame()
ls=matrix(rep(NA,9*44),nrow=44)
for (p in p1:p2)
{
for(q in q1:q2)
{
for(P in P1:P2)
{
for(Q in Q1:Q2)
{
for(d in d1:d2)
{
if(p+d+q+P+D+Q<=8)
{
model<- Arima(data,order=c(p-1,d,q-1),seasonal=c(P-1,D,Q-1))
ls[i,]= c(p-1,d,q-1,P-1,D,Q-1,model$aic,model$bic,model$aicc)
i=i+1
#print(i)
}
}
}
}
}
}
temp= as.data.frame(ls)
names(temp)= c("p","d","q","P","D","Q","AIC","BIC","AICc")
temp
}
##q=1,2 Q=0 , p=1,2 P=0 d=0
output=SARIMA.c(p1=1,p2=3,q1=1,q2=4,P1=1,P2=3,Q1=1,Q2=2,d1=0,d2=0,data=res.fit)
output p d q P D Q AIC BIC AICc
1 0 0 0 0 0 0 4947.866 4956.796 4947.885
2 0 0 0 0 0 1 4947.866 4956.796 4947.885
3 0 0 0 1 0 0 4947.866 4956.796 4947.885
4 0 0 0 1 0 1 4947.866 4956.796 4947.885
5 0 0 0 2 0 0 4947.866 4956.796 4947.885
6 0 0 0 2 0 1 4947.866 4956.796 4947.885
7 0 0 1 0 0 0 4496.989 4510.383 4497.027
8 0 0 1 0 0 1 4496.989 4510.383 4497.027
9 0 0 1 1 0 0 4496.989 4510.383 4497.027
10 0 0 1 1 0 1 4496.989 4510.383 4497.027
11 0 0 1 2 0 0 4496.989 4510.383 4497.027
12 0 0 1 2 0 1 4496.989 4510.383 4497.027
13 0 0 2 0 0 0 4203.872 4221.731 4203.935
14 0 0 2 0 0 1 4203.872 4221.731 4203.935
15 0 0 2 1 0 0 4203.872 4221.731 4203.935
16 0 0 2 1 0 1 4203.872 4221.731 4203.935
17 0 0 2 2 0 0 4203.872 4221.731 4203.935
18 0 0 3 0 0 0 4039.712 4062.035 4039.807
19 0 0 3 0 0 1 4039.712 4062.035 4039.807
20 0 0 3 1 0 0 4039.712 4062.035 4039.807
21 1 0 0 0 0 0 3812.180 3825.574 3812.218
22 1 0 0 0 0 1 3812.180 3825.574 3812.218
23 1 0 0 1 0 0 3812.180 3825.574 3812.218
24 1 0 0 1 0 1 3812.180 3825.574 3812.218
25 1 0 0 2 0 0 3812.180 3825.574 3812.218
26 1 0 0 2 0 1 3812.180 3825.574 3812.218
27 1 0 1 0 0 0 3714.614 3732.473 3714.677
28 1 0 1 0 0 1 3714.614 3732.473 3714.677
29 1 0 1 1 0 0 3714.614 3732.473 3714.677
30 1 0 1 1 0 1 3714.614 3732.473 3714.677
31 1 0 1 2 0 0 3714.614 3732.473 3714.677
32 1 0 2 0 0 0 3712.704 3735.027 3712.799
33 1 0 2 0 0 1 3712.704 3735.027 3712.799
34 1 0 2 1 0 0 3712.704 3735.027 3712.799
35 1 0 3 0 0 0 3692.626 3719.414 3692.758
36 2 0 0 0 0 0 3703.861 3721.719 3703.924
37 2 0 0 0 0 1 3703.861 3721.719 3703.924
38 2 0 0 1 0 0 3703.861 3721.719 3703.924
39 2 0 0 1 0 1 3703.861 3721.719 3703.924
40 2 0 0 2 0 0 3703.861 3721.719 3703.924
41 2 0 1 0 0 0 3705.857 3728.180 3705.952
42 2 0 1 0 0 1 3705.857 3728.180 3705.952
43 2 0 1 1 0 0 3705.857 3728.180 3705.952
44 2 0 2 0 0 0 3707.542 3734.330 3707.675output[which.min(output$AIC),] p d q P D Q AIC BIC AICc
35 1 0 3 0 0 0 3692.626 3719.414 3692.758output[which.min(output$BIC),] p d q P D Q AIC BIC AICc
35 1 0 3 0 0 0 3692.626 3719.414 3692.758output[which.min(output$AICc),] p d q P D Q AIC BIC AICc
35 1 0 3 0 0 0 3692.626 3719.414 3692.758model_outputma2 <- capture.output(sarima(res.fit, 1,0,3,0,0,0))
cat(model_outputma2[57:89], model_outputma2[length(model_outputar2)], sep = "\n")$fit
Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
xreg = xmean, include.mean = FALSE, transform.pars = trans, fixed = fixed,
optim.control = list(trace = trc, REPORT = 1, reltol = tol))
Coefficients:
ar1 ma1 ma2 ma3 xmean
0.9771 -0.4537 0.1864 -0.196 -0.2391
s.e. 0.0091 0.0402 0.0458 0.041 3.6951
sigma^2 estimated as 18.02: log likelihood = -1840.31, aic = 3692.63
$degrees_of_freedom
[1] 637
$ttable
Estimate SE t.value p.value
ar1 0.9771 0.0091 107.5191 0.0000
ma1 -0.4537 0.0402 -11.2978 0.0000
ma2 0.1864 0.0458 4.0690 0.0001
ma3 -0.1960 0.0410 -4.7836 0.0000
xmean -0.2391 3.6951 -0.0647 0.9484
$AIC
[1] 5.751754
$AICc
[1] 5.751901
$BIC
[1] 5.793479
NAarimaModel_1 <- arima(res.fit, order = c(1,0,3))
forecast1=predict(arimaModel_1, 5)
# create df with fcast preds and +-1.96 SE for 95% CI Bands
farimax_df <- data.frame(
Year = 2021:2025,
fcst = as.numeric(forecast1$pred),
lower = as.numeric(forecast1$pred - 1.96 * forecast1$se),
upper = as.numeric(forecast1$pred + 1.96 * forecast1$se)
)
#plot(forecast1$pred, main = "ARIMA(2,0,0) Forecast For 5 Years", xlab = "Time", ylab = "Values", col = "red")
(arimax_plot <- ggplot(data=farimax_df, aes(x=Year, y=fcst, group = 1)) +
geom_line(aes(color="Forecast"), linewidth=1) +
geom_ribbon(aes(ymin=lower, ymax=upper, fill="Confidence Interval"), alpha=0.1) +
labs(title="ARIMA(1,0,3) Forecasts for CPI from 2021 to 2025",
y="CPI",
color="", fill="",
caption="Data Sources: FRED") +
scale_color_manual(values = c("Forecast"="red")) +
scale_fill_manual(values = c("95% Confidence Interval"="steelblue")) +
theme_minimal() +
theme(plot.caption.position = "plot", plot.caption = element_text(size=8)))
Final Results: Unveiling the Impact of Post-COVID Data on Economic Forecasting: Insights from VAR(2) and ARIMAX Models
The Emergence of VAR(2) as the Front-Runner in Post-Pandemic Economic Forecasting In the realm of economic forecasting, the COVID-19 pandemic has catalyzed a paradigm shift, challenging conventional models and necessitating a reevaluation of data relevance. Our analysis reveals a compelling narrative: the Vector Autoregression (VAR) model, specifically VAR(2), emerges as a beacon of adaptability and accuracy in this new economic era.
Key Findings:
VAR(2) Model Proficiency: The VAR(2) model, utilizing data exclusively from the post-COVID era, significantly outstripped other models in forecasting accuracy. This model adeptly captured the increasing economic trends unique to this period, underscoring its ability to adapt to rapid changes.
Post-COVID Data’s Preeminence: The analysis underscores the importance of focusing on recent, post-pandemic data. Traditional datasets, encompassing years prior to COVID-19, now offer less predictive value for current and future economic states. This shift is vividly reflected in the performance leap of the VAR(2) model, which thrives on post-COVID data.
Comparing VAR(2) with ARIMAX Models: A Tale of Two Methodologies
Our journey into economic forecasting doesn’t end with VAR models. A comparison with ARIMAX (Autoregressive Integrated Moving Average with Exogenous Variables) models painted a nuanced picture:
ARIMAX Model Disparities: Models trained on all-inclusive data spanning from 1970 to 2020 paled in comparison to those trained solely on post-COVID data. This was particularly evident in the leap in adjusted R-squared values, indicating a more accurate fit when focusing on recent data.
VAR’s Superiority: Despite the improvements seen in ARIMAX models using post-COVID data, they fell short of the predictive prowess exhibited by VAR models. This highlighted VAR’s superior ability to interpret and forecast based on the interplay between multiple economic indicators.
Deciphering the Economic Narrative: The Role of Key Variables
Our exploration revealed two variables as particularly influential in forecasting the yearly Consumer Price Index (CPI) in the US: Personal Consumption and Treasury Yield. These variables showcased a high correlation with the CPI, hinting at their pivotal roles in the post-pandemic economic landscape.
Personal Consumption: Reflecting shifts in consumer behavior during the pandemic, this variable has become a critical indicator of economic health.
Treasury Yield: Serving as a barometer for broader economic trends and policies, Treasury Yield’s correlation with CPI underscores its significance in our current economic context.
The Road Ahead: Embracing Change in Economic
Forecasting In conclusion, our analysis not only highlights the VAR(2) model’s adeptness in navigating the post-COVID economic waters but also underscores the imperative to prioritize recent data for more accurate forecasting. The pandemic has irrevocably altered the economic fabric, mandating an adaptive approach to forecasting models and data selection.
Stay tuned as we continue to explore and unravel the complexities of economic forecasting in this ever-changing world.